Consider the following statements:
S_1: There exist infinite sets A, B, C such that A ∩ (B ∪ C) is finite.
S_2: There exist two irrational numbers x and y such that (x + y) is rational.
Which of the following is true about S_1 and S_2?
A. Only S_1 is correct
B. Only S_2 is correct
C. Both S_1 and S_2 are correct
D. None of S_1 and S_2 is correct
GATE 2001 · Discrete Mathematics · Set Theory · medium
Answer: Both S_1 and S_2 are correct — answer C.
Verify S_1 with a concrete example: Let A = N (all natural numbers), B = {0}, C = {1}. All three sets are infinite (N has infinitely many elements; single-element sets are finite — wait, we need B and C infinite too). Correct witness: A = N, B = even numbers, C = {1}. Then B ∪ C contains 1 and all evens. A ∩ (B ∪ C) = evens ∪ {1} — still infinite. Better: A = N, B = {2, 4, 6, ...}, C = {3} is not fully infinite for C. Use A = N, B = N, C = N; A ∩ (B ∪ C) = N (infinite). Instead: let A = {2, 4, 6, ...}, B = {1, 2, 3}, C = {5, 6, 7, ...} (both infinite). A ∩ (B ∪ C) = {2} ∪ {6,8,10,...} still infinite. Simplest valid witness: A = {1,2,3,...} = N, B = {-1,-2,-3,...}, C = {-1,-2,-3,...}: A ∩ (B ∪ C) = empty set (finite!), and A, B, C are all infinite. So S_1 is TRUE.
Verify S_2 with a concrete example: Both sqrt(2) and -sqrt(2) are irrational (if -sqrt(2) = p/q then sqrt(2) = -p/q, rational — contradiction). Their sum is sqrt(2) + (-sqrt(2)) = 0, which is rational. So S_2 is TRUE.